## Papers and Preprints

### List of Publications

I no longer actively maintain a list of publications on this page; instead my list of recent publications and preprints from arxiv.org is shown below automatically, courtesy of the myarticles widget. You can also use the following resources:
• My ORCID record. (The open and non-profit ORCID system provides a unique researcher ID that can be associated to articles and is highly recommended.)
• Preprint versions of all my papers on arxiv.org.
• You can find a list of my publications, and associated reviews, on MathSciNet if you are connecting from a subscribing institution.
• My (only occasionally maintained) Google Scholar Profile.
• A Wiki on which I will aim to maintain errata and commentary for some of my published and forthcoming papers.

See below in addition for Preprints, Theses and Books that do not appear on the above lists, as well as a (no longer maintained) list of previous errata.

Current (August 2015) list of co-authors (completed papers only): Alhabib, Barański, Bergweiler, Epstein, Förster, Jarque, Fagella, Kahn, Lyubich, Mayer, Mihaljević-Brandt, Rippon, Rottenfußer, Rückert, Schleicher, Shen, Sixsmith, Stallard, Urbański, van Strien.

### Summaries and errata

The goal of this section was to make available some summaries of what is contained in some of my papers, aimed at other experts in the field, as well as to collect misprints and errata. The section is not being maintain; instead any such information will in the future be updated on this Wiki. If you find any errors in any papers, please e-mail me and I will include an erratum there!

This information is currently very incomplete, but I will try to add to it as time goes by.

#### Topological dynamics of exponential maps on their escaping sets

Ergodic Theory Dynam. Systems 26 (2006), no. 6, 1939 - 1975. arXiv:math.DS/0309107; published version.

Errata:

• In the proof of Theorem 1.1, on page 1950, the number R should be defined as R:=exp(Q(K)+1)+K, instead of R:=Q(K)+1. (Many thanks to Anup Raja Lamichhane for pointing out this error.)

#### Prime ends and local connectivity

Bull. London Math. Soc. 40 (2008), no. 4, 817 - 826. arxiv:math.GN/0309022. Published version: Abstract, PDF.

Note (March 2012):Donald Sarason has kindly pointed out that Theorem 1.1 was proved by Marie Torhorst in 1918 in her dissertation (Über die Randmenge einfach-zusammenhängender ebener Gebiete); the result was published in Math. Z. in 1921. It appears that this result had largely been forgotten: I could not find a single reference to Torhorst on MathSciNet, and no-one I spoke to about my paper while I was preparing it was aware of it either!

Moreover, Don himself wrote a paper with the exact same title as mine in the 1960s, which reproves Torhorst's result from the work of Ursell and Young, which is the same argument by which Theorem 1.1 is established in my paper. However, his paper was not accepted for publication at the time, as he explains:

I submitted the paper to the Michigan Math. J., then edited by George Piranian, the person who taught me about prime ends and much more about complex analysis. (George is one of my mathematical heroes.) George discussed the paper with Collingwood, one of his collaborators. Their conclusion was that interest in prime ends at the time was at such a low ebb that the paper was likely to be largely ignored.

I did publish an abstract of the paper in the Notices of the A.M.S. (Vol 16 (1969), p. 701). At the time the Notices published abstracts of talks given at society meetings, plus what I think were called by-title abstracts, which any member of the society could use to announce a result. If my memory is correct, I received as a result of the abstract only one request for a copy of the paper.

Don's 1960s manuscript, along with George Piranian's letter and the announcement in the Notices, are contained in this PDF file, which he has kindly allowed me to make available.

To my knowledge, Theorem 1.3, which a characterization of local connectivity at a point and from which Theorem 1.1 follows using the Ursell-Young result, has not previously appeared elsewhere. (Note, however, that the argument that proves the "only if" direction is the same as the one that appears already in Don's paper, which also contains the "if" direction in the special case that every prime end whose impression contains the point in question is of the first kind.)

#### Hyperbolic dimension and radial Julia sets of transcendental functions

Proc. Amer. Math. Soc. 137 (2009), 1411-1420. arXiv:0712.4267; published version.

Errata:

• The reference [O] should have listed Adam Epstein and Richard Oudkerk as authors, and should have been denoted by [EO].

#### Dynamic rays of bounded-type entire functions

With Günter Rottenfußer, Johannes Rückert and Dierk Schleicher.

Ann. of Math. 173 (2011), no. 1, 77-125. arxiv:0704.3213; published version.

Errata:

• In part (c) of the statement of Proposition 5.4, M should be M' (twice).
• In the statement of Lemma 5.7 (Linear head-start is preserved by composition), "F_1 has bounded slope and all F_i satisfy uniform linear head-start conditions" should be replaced by "All F_i have bounded slope and satisfy uniform linear head-start conditions." The second paragraph of the proof should be replaced by the following:

Let $\alpha$ and $\beta$ be such that ${F}_{i}\in {ℬ}_{\mathrm{log}}^{n}\left(\alpha ,\beta \right)$ for all $i$ . For $i=1,\dots ,n$ , let ${K}_{i}$ and ${M}_{i}\text{'}$ be the constants from Proposition 5.4 (c), applied to ${F}_{i}$ . We set $K:=\underset{i}{\mathrm{max}}{K}_{i}$ and $M:=\mathrm{max}\left\{\delta ,\underset{i}{\mathrm{max}}{M}_{i}\text{'}\right\}$ , where $\delta =\delta \left(\alpha ,\beta ,K,0\right)$ is the constant from Lemma 5.2. Now fix $i$ and let $T$ be a tract of ${F}_{i}$ . Let $w,z\in T$ such that $\mathrm{Re}w>K\mathrm{Re}z+M$ and such that ${F}_{i}\left(z\right)$ and ${F}_{i}\left(w\right)$ belong to the same tract of ${F}_{i+1}$ (where we use the convention that ${F}_{n+1}={F}_{1}$ ). Then $|w-z|\ge \mathrm{Re}w-\mathrm{Re}z>M\ge \delta$ , and Lemma 5.2 gives that  $\mathrm{Re}{F}_{i}\left(w\right)>K\mathrm{Re}{F}_{i}\left(z\right)+M\phantom{\rule{1.00em}{0ex}}\text{or}\phantom{\rule{1.00em}{0ex}}\mathrm{Re}{F}_{i}\left(z\right)>K\mathrm{Re}{F}_{i}\left(w\right)+M\phantom{\rule{0.278em}{0ex}}.$ ()
By Proposition 5.4 (c), the first inequality must hold. It follows that ${G}_{a}$ satisfies a uniform linear head-start condition with constants $K$ and $M$ .

(Many thanks to Sebastian Vogel for pointing out this error.)

#### Exotic Baker and wandering domains of Ahlfors islands functions

With Phil Rippon (Journal d'Analyse Mathematique)

This paper constructs examples of functions satisfying Adam Epstein's "Ahlfors islands property" (see "Hyperbolic dimension and radial Julia sets", above) which have various interesting dynamical or function-theoretic properties, using approximation theory.

If X is the Riemann sphere or a torus, and W is a proper subdomain of X, then we show that there are Ahlfors islands maps g:W->X. Moreover, these can be chosen such that:

a) g has a Baker domain whose set of limit functions coincides with any prescribed compact connected subset of the boundary of W that can be written as the accumulation set of a curve in W; moreover the iterates in the Baker domain can be chosen to tend to infinity arbitrarily slowly;
b) g has a wandering domain whose set of limit functions coincides with any prescribed compact subset of the boundary of W;
c) given a prescribed C1 curve gamma tending to the boundary, g can be constructed with a logarithmic asymptotic value that has gamma as an asymptotic curve.

Note that c) is a nondynamical statement, and in fact W can be replaced by any proper subdomain of any compact Riemann surface Y. In particular, it follows that there are no restrictions on the possible domains of Ahlfors islands functions when the target Riemann surface has genus at most 1.

While most of the results are well-known when X is the Riemann sphere and W is the complex plane or the punctured plane, the statement about escape speeds in a) also yields an apparently new result for entire functions: The rate of escape in a Baker domain of a transcendental entire function can be arbitrarily slow.

The construction uses Arakelian's approximation theorem and its generalization to analytic functions on Riemann surfaces due to Scheinberg. In order to be able to construct Baker domains, we need to establish approximation results for functions taking values in a simply-connected domain that are a priori stronger than those provided by the Arakelian-Scheinberg theorem.

We also construct examples as in a), b) and c) above when X is a compact hyperbolic Riemann surface. However, in this case the constructed function will be an Ahlfors islands map g:W'->X with the stated properties, where W' is a proper subdomain of W.